MATH 428: Topics in Complex Analysis, Fall 98

Description:

Weierstrass's work on the analytic continuation of holomorphic functions led Riemann to develop the notion of the Riemann surface  as a natural domain for an algebraic function.  Here an algebraic function is a (possibly  multiple-valued) solution  w = w(z)  of an equation  a0(z)wn + a1(z)wn-1 +  ... + an(z) = 0  were the coefficients  aj(z)  are  polynomials in  z .  In more recent terminology, a Riemann surface is understood to be a 1-dimensional complex manifold or equivalently a real 2 dimensional manifold with a distinguished collection of coordinate charts whose transition functions preserve both angles and orientation. A fundamental theorem of Riemann is that any compact Riemann surface locally so defined corresponds globally to an algebraic function as above.  It is well-known that Riemann surface theory is not only a point of departure but even at the historical roots of a large part of  mathematics, in particular, topology, differential geometry, algebraic geometry, algebraic number theory, and partial differential equations. While we cannot hope to make all these connections, our present wish list includes the following topics:
  • Examples; sphere, tori, branched covers
  • Integration of rational functions, topology of surfaces, Riemann-Hurwitz formula
  • Differential forms, Weyl's lemma, existence of meromorphic functions and forms
  • Riemann existence theorem, Riemann- Roch theorem, Abel's theorem, Jacobi Inversion
  • Uniformization and classification

    • Meets:

    9:25-10:40 TuTh in Herman Brown 423

    Instructor:

    Robert Hardt
    Office: Herman Brown 430; Office hours: 1-2 MTuW (and others by appt.)

    Email: hardt@rice.edu
    Telephone: ext 3280

    Text:

    Harvey Cohn, Conformal Mapping on Riemann Surfaces, Dover, 1995.
    This 1967 book gives many exercises and a careful, somewhat old-fashioned, introduction to the most basic topics in classical Riemann surface theory.

    Two other useful references are the paper of H.L.Royden, Function theory on compact Riemann surfaces, J. D'analyse Math. 8(1967), 295-327 (which we will follow) and the book by H. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, 1991 (which includes everything we will cover and much more).

    Readings:

    Cohn: Sec.3.1, 3.3, 3.4
    Cohn: Sec.2.1, 2.1, 2.3, 2.4
    Cohn: Sec.7.3, 5.3, 8.1, 8.2, 8.3
     

    Announcements:

    1. Thursday, October 1, while I am away, Professor J. Anderson will lead the class with a discussion of Mobius transformations. This subject, which is related to Professor Anderson's research, is treated in Cohn, sec.2.5-2.7.

    2. See below the corrections for  Assignment 5.

    3. Below are some more hints on some of the older problems. If you didn't get them the first time,
    you can try them again and hand in for some extra credit.

    4. Look at the list of misprints, etc. in the Royden paper. Please e-mail any you find.

    5. Correction for Nov. 17 lecture.  In the proof of the existence of a nonconstant meromorphic function, we used a holomorphic 1 form   a .
    However, we didn't guarantee that this was not identically 0 . One should use alternately a meromorphic 1 form  a  which is regular at  p0 (the pole of
    b ) and has an order 2 pole at another point  q0 .  Then  b /a  is the desired meromorphic function.

    6. A 4-hour open-book, open-notes, take-home final exam will be distributed on the last day of classes. It will be due at the end of exam period.  (Saturday, Dec.19, 4pm). Late homework will be accepted then also. However, you should try to finish this as soon as possible as much of the
    exam will be similar to the homework. I will be around during this whole period to answer questions.

    7. Please pick up the Xeroxed materials from Royden and Farkas-Kra. You may also pick up the notes on Algebraic Riemann Surfaces or click here.

    8. Also please pick up  the notes on Riemann-Roch or click here.

    9. For solutions to the final exam click here.

    Homepage:

    http://math.rice.edu/~hardt/428

    Grading:

    The final grade for the course will be determined as follows: 
                  Homework        60%
             
              Final exam      40%

    Homework:

    There will be a homework assignment approximately each week. All homework is usually due each Thursday, one week after it is assigned. The homework is not pledged. You are encouraged to discuss the homework, and to work together on the problems. However each student is responsible for the final preparation of his or her own homework papers.

     I will be adding hints on this homepage to problems even after they are due. If you don't get a problem the first time, check the hint and give it another try. Then hand it in for some partial credit.
     

    Assignment 1 (now due 9/15):

    Cohn: sec.8.3#5,#6, sec.2.2#1,#2, and Prove: Any holomorphic function on a compact Riemann surface is constant.
    (Hint: A continuous function on a compact space attains its maximum value at some point. Apply the maximum principle to the holophorphic function in a coordinate neighborhood of such a point.)

    Remarks.  I should have translated some of the old-fashioned language of Cohn.

        In 8.3#6, biunique means bijective.
        Dihedral angle refers in general to the angle between two planes. In 2.2#2 the dihedral angle through PQ refers to two planes both passing through P and Q. Note that these intersect the sphere along 2 curves whose stereographic images are the two lines in the horizontal plane obtained by intersecting with the two planes. The tangent lines of these 2 curves at P are the intersections of the tangent plane at P with these two planes.

    Assignment 2 (due 9/22):

    Cohn: sec.2.3#2, sec.2.4#5, sec.3.3#2
    Also:  Find a nonconstant meromorphic function on the standard torus. (Hint: This corresponds to finding a doubly-periodic meromorphic function on the complex plane.)
    (Hint: Try Sm,n(z-m-ni)-k.  How large do you need to choose k to make this converge?)

    Assignment 3 (due 9/29):
     TWO TORI :
    Suppose T = C / ~ where ~ is the equivalence relation on the complex plane C given by
                 z ~ w  if and only if  z - w = m + ni  for some integers m, n .
    Ex.1. Show that  T  is homeomorphic to  { (z,w) in CxC : |z|=|w|=1 } by finding a specific bijective continuous map between the two sets.
    Ex. 2. Show that T has the structure of a Riemann surface so that the projection map taking a complex number z to its ~ equivalence class is holomorphic. (In fact an infinite-sheeted holomorphic covering map.)
    Ex. 3. Show this projection map is bijective on the open unit square {s + t i : 0<s<1 , 0<t<1}.
    Suppose S = C / # where # is the equivalence relation on the complex plane C given by
               z # w  if and only if  z - w = m + n(1/2 + i) for some integers m, n .
    Ex. 4. Show that  S  is homeomorphic to  { (z,w) in CxC : |z|=|w|=1 } by finding a specific bijective continuous map between the two sets.
    Ex. 5. Show that S has the structure of a Riemann surface so that the projection map taking a complex number z to its # equivalence class is holomorphic. (In fact an infinite-sheeted holomorphic covering map.)
    Ex. 6. Show this projection map is bijective on the open parallelogram {s + t(1/2 + i) : 0<s<1 , 0<t<1}
    Ex. 7. Show that there is no biholomorphic map between  T  and  S . [Hint: Show how such a map must lift to a holomorphic map f : C -> C . Examine f(z+1) , f(z+i) , and the behavior of  f  at oo .]
    (Alternate Hint: Show that f ' must be a bounded entire function, hence f(z) = az+b for some complex a,b.)

    Assignment 4 (due 10/8):

    Cohn: sec.5.3#6,#8 (see the last 3 paragraphs of sec.4.3)
    Ex. 3. Describe all degree 1 holomorphic maps from the standard torus to itself.

    Ex. 4.  Find an unbranched holomorphic map from the standard torus to itself with degree > 1.

    Ex. 5. Does there exist an unbranched holomorphic map from the standard torus to itself with degree 2 ? If yes,
    find one. If not, explain.

    (Hint for Ex.3,4,5. Use the hint from the above Ex.7.  Then find the possible a, b.)

    Assignment 5 (now due 10/22):

    Corrected Ex. 1.  Here we define as in class ("round d") by  df = f z dz  and  ("round d bar") be  /d f = f /z d /z
    for smooth functions  f  and  d w = v z dz d /z  and  /d w = -u /z dz d /z  for 1 forms  w = u dz + v d /z .
    (Sorry I couldn't find more math symbols in the available fonts.)
    Show that for any smooth function  f  on has:  d2f = 0 , (/d)2f = 0, (d/d + /dd)f = 0 .

    Corrected Ex. 2.  Verify the relations for a smooth function f  and a smooth 1 form w :
    df = (1/2)(d + i*d)f ,  (/d)f = (1/2)(d - i*d)f ,  dw= (1/2)(d - id*)w, (/d)w= (1/2)(d + id*)w, (d/d)f = -(/dd)f = (1/2i)d*df

    Cohn: sec.8.1,#1, sec.8.3#3, #9 .

    Remarks on 8.3,#3.  Cohn's notation  dy* corresponds to our notation *dy. For y being a harmonic function (which is the situation here) y* can also be understood to be a conjugate harmonic function
    of y .  Since any two harmonic conjugates of a given harmonic function differ by a constant, the 1 form  dy* is well-defined.  The question concerns finding different  a  so that the real and imaginary parts of the
    (multi-valued) holomorphic function a log z  either are both, neither, or mixed single-valued.

    Assignment 6 (now due 10/29): Click here. (Check Ex.4. There was a sign error in the earlier version.See the additional hints.)

    Assignment 7 (due 11/5) : Click here.

    Assignment 8 (due 11/12) :
    1. On the Riemann sphere (the extended complex plane), find the Green's function  g(z , 0 ,  i  , -i ) .

    2.  Recall that the flux of  u  is the integral of  *du .  Find the flux (possibly complex-valued) across the unit circle of

                                 a.  u(z) = log |z|               b.  u(z) = z           c.  1 / z 2               d.  u(z) = 1 / z

    Hint:  For b, c, and d, the formula   *d = (1 / i) (d - /d)  may be useful.

    3.  Show how the quotient of  2 nonconstant meromorphic 1 forms gives a meromorphic function.
    (i.e. Show that if  a = f(z)dz  and  b=g(z)dz  in a local coordinate  z ,  then  f /g  is well-defined independent of the local coordinate.)

    4. On the standard torus  T = { (z,w) in CxC : |z|=|w|=1 } , find the Weyl decomposition of the 1 form

                 (3 + cos 2q   - sin 2q ) dq - (1 - cos 4q) df

    where  q  = arg z  and f = arg w .

    5. Show that the 4 terms in the complex decomposition (from lecture) of a smooth 1 form

                            a  =   d f1 +  /d f2  +  w1  +  w2

    are mutually orthogonal in L2 .

    Assignment 9 (due 11/19) :

    1.  Find the complex Weyl decomposition of the 1 form from exercise #4 of Assignment 8.

    For problems 2-5 let  M  be a compact Riemann surfaces with  N  components.

    2.  Let  H0(M) be the real vectorspace of smooth closed  0  forms on  M .  Show that  dim H0(M) = N  by finding a basis.

    3.  Let   Z1(M) be the real vectorspace of smooth closed  1  forms on  M ,  and B1(M) be the real vectorspace of smooth exact  1  forms on  M .  Show that  dim Z1(M) = oo = dim B1(M) .

    4.  Show that every element of  the quotient space H1(M) =  Z1(M) / B1(M)  may be represented by a unique harmonic 1 form .

    5.  Let   Z2(M) be the real vectorspace of smooth  2  forms on  M ,  B2(M) be the real vectorspace of smooth exact  2  forms on  M , and   H2(M) =  Z2(M) / B2(M) .  Show that  dim H2(M) = N .

    Assignment 10 (due 11/31) :

    1. What is the dimension of the space of meromorphic 1 forms on a compact Riemann surface of genus 2 ?

    2. On the standard torus  T = { (z,w) in CxC : |z|=|w|=1 } , find bases for the spaces of
     (a) harmonic 1 forms on  T   (b) holomorphic 1 forms on  T   (c) anti-holomorphic 1 forms on  T

    3. Using the standard model for a genus g oriented surface, obtain a triangulation, and show that the Euler characteristic
    (i.e. #(vertices) - #(edges) + #(triangles) ) equals  2 - 2 g .

    Assignment 11 (due 12/8) :

    1. In contrast to Problem 1 of the previous assignment:
    What is the (real) dimension of the space of harmonic functions on a compact Riemann surface of genus 2 ?
        What is the (complex) dimension of the space of holomorphic functions on a compact Riemann surface of genus 2 ?
    What is the (real) dimension of the space of harmonic 1 forms on a compact Riemann surface of genus 2 ?
        What is the (complex) dimension of the space of holomorphic 1 forms on a compact Riemann surface of genus 2 ?

    2. Describe topologically {(z,w) in C x C :  (z2-9)(w2 - z5 = 0)(w-2z) = 0 }.

    3. Show (using Riemann-Roch) that a meromorphic function on a torus cannot have a single simple pole.
     
     
     
     

    This page is maintained by Robert Hardt ( email )
    Last edited 12/9/98.